[Math] Intersection of parabola and circle

Question

Is it possible to place circle and a parabola on the plane so that their intersection consist of exactly two points, one point being a point of tangency, and the other point a transversal intersection?

Answer 1

Let the circle be the unit circle, and take the parabola to have the equation $y-k=a(x-h)^2$, with $a\neq 0$. Note that the slope of the line tangent to the parabola at $(x,y)$ is given by $m = 2 a (x-h)$.

Let $T(\cos\theta,\sin\theta)$ be the point of tangency. (We may assume throughout that $\sin\theta \neq 0$, since our (presumably non-degenerate) parabola has no vertical tangents. Note also that distinct points of intersection must correspond to distinct $x$ coordinates.) Then, as $T$ must satisfy the parabola equation, we have $$\sin\theta - k = a ( \cos\theta - h )^2 \qquad (1)$$ Now, the line tangent to the circle at $T$ necessarily has slope $-\cot\theta$; equating this with the slope relative to the parabola gives $$\cos\theta = -2 a \sin\theta (\cos\theta - h ) \qquad (2)$$

We can solve $(2)$ for $h$, and then $(1)$ for $k$:

$$ h = \frac{\cos\theta \left( 1 + 2 a \sin\theta \right)}{2a\sin\theta} \qquad k = \frac{ 4 a \sin^3\theta - \cos^2\theta }{4a\sin\theta^2}$$ whence the parabola equation becomes

$$y \sin\theta = a x^2 \sin\theta - x \cos\theta \left( 1 + 2 a \sin\theta \right) + 1 + a \cos^2\theta \sin\theta $$

Using that equation to eliminate $y$ in $x^2+y^2=1$ gives this polynomial equation in $x$:

$$\left( x - \cos\theta \right)^2 \left( \left( a \left( x - \cos\theta \right)\sin\theta - \cos\theta\right)^2 + \sin^2\theta \left( 1 + 2 a \sin\theta \right)\right) = 0 \qquad (3) $$

For the second factor of $(3)$ to have any roots requires the quantity $b := 1+2a\sin\theta$ to be non-positive. If $b$ vanishes, we get the double root $x=-\cos\theta$ (the mirror image of $x=\cos\theta$) corresponding to a second point of circle-parabola tangency. For us to have a point of non-tangent intersection, we require $b$ to be strictly negative, and $x=\cos\theta$ to be a root of that second factor. The latter condition implies, by substitution: $$2a\sin^3\theta=-1$$ giving $b=-\cot^2\theta$, which is strictly negative (and defined) for $\theta \neq n \pi/2$. Moreover, equation $(3)$ reduces nicely to $$0 = (x-\cos\theta)^3\left(x-4\cos^3\theta+3\cos\theta\right) = (x-\cos\theta)^3(x-\cos 3\theta)$$ providing the root $x=\cos3\theta$. Double-checking with the parabola equation, we see that the corresponding $y$ value should be $-\sin3\theta$; thus, we can write the second point of intersection as $S(\cos(-3\theta), \sin(-3\theta))$.

The final parabola equation itself is $$ 2 y \sin^3\theta = - \left( x - \cos^3\theta \right)^2 + \sin^4\theta\left( 3 - \sin^2\theta \right)$$

Here's an animation:

(Animation, and interstitial symbol-crunching, courtesy of Mathematica.)